Probability

The One Thing

Probability measures how likely an event is, using values from 0 to 1 or from 0% to 100%. Combined events are analysed by identifying the relevant sample space and determining whether events are mutually exclusive, independent or dependent before applying an appropriate rule.

Definitions and Results

• Probability: A measure of how likely an event is to happen. It may be written as a fraction, decimal or percentage.
• Probability scale: A scale from 0 to 1, where 0 means impossible and 1 means certain. The equivalent percentage scale runs from 0% to 100%.
• Outcome: A possible result of an experiment, such as rolling a 4 on a die.
• Event: A single outcome or collection of outcomes satisfying a condition.
• Sample space: The complete list of all possible outcomes of an experiment.
• Complement: The event that an event does not happen. If the probability of $A$ is $P(A)$, then

$$P(\text{not }A)=1-P(A).$$

• Mutually exclusive events: Events that cannot happen at the same time, such as rolling an even number and an odd number on one roll.
• Independent events: Events where the outcome of one event does not affect the probability of the other event.
• Dependent events: Events where the outcome of one event changes the probability of the next event, often because items are not replaced.
• Experimental probability: The probability estimated from observed results:

$$\text{Experimental probability}=\frac{\text{number of successful trials}}{\text{total number of trials}}.$$

• Theoretical probability: The probability calculated from equally likely outcomes or a mathematical model.
• Relative frequency: The proportion of trials producing a particular outcome:

$$\text{Relative frequency}=\frac{\text{frequency}}{\text{total frequency}}.$$

• Expected frequency: The predicted number of times an event should occur:

$$\text{Expected frequency}=\text{probability}\times \text{number of trials}.$$

• Tree diagram: A diagram showing possible sequences of events, with probabilities written on branches.
• Sampling: Selecting part of a population to collect information about the whole population.
• Population: The complete group being studied.
• Sample: A smaller group selected from the population.
• Bias: A systematic feature of a method that makes a sample or result unrepresentative.
• Random sample: A sample in which every member of the population has an equal chance of being selected.

Important results and conditions:

• Probability values satisfy

$$0\le P(A)\le 1.$$

• For an impossible event,

$$P(A)=0.$$

• For a certain event,

$$P(A)=1.$$

• For equally likely outcomes,

$$P(A)=\frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}.$$

• The probabilities of all outcomes in a complete sample space add to 1.
• For mutually exclusive events,

$$P(A\text{ or }B)=P(A)+P(B).$$

• For events that are not mutually exclusive,

$$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B).$$

• For independent events,

$$P(A\text{ and }B)=P(A)\times P(B).$$

• In a tree diagram, probabilities along a sequence are multiplied.
• Probabilities for alternative branches representing the same final event are added.
• If items are selected without replacement, probabilities may change after each selection.
• As the number of trials increases, experimental probability often gets closer to theoretical probability, although results can still vary.
• A sample should be sufficiently large and representative to support reliable conclusions.
• Random sampling reduces selection bias.
• Systematic sampling selects members using a regular interval.
• Stratified sampling reflects important subgroups in the population.
• A sample may be unrepresentative if it is too small, selected from only one subgroup, affected by voluntary response, or based on leading questions.
• A probability between 0 and 1 describes likelihood but does not guarantee the result of one individual trial.
• Theoretical probability depends on a mathematical model and its assumptions, whereas experimental probability depends on collected data.

Worked Methods

Calculating the probability of an event

• List the complete sample space.
• Count the favourable outcomes.
• Count the total possible outcomes.
• Apply

$$P(A)=\frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}.$$

• Express the result as a fraction, decimal or percentage if required.

For example, rolling a 4 on a die is an outcome. The event could consist of one or more outcomes satisfying a stated condition.

Using complements

• Identify the event $A$.
• Identify the complementary event, written as “not $A$”.
• Apply

$$P(\text{not }A)=1-P(A).$$

This is useful when calculating the probability that an event does not happen.

Combining mutually exclusive events

• Check that the events cannot occur at the same time.
• Add their probabilities:

$$P(A\text{ or }B)=P(A)+P(B).$$

Rolling an even number and rolling an odd number on one roll are mutually exclusive events, because one roll cannot produce both an even and an odd result.

Combining events that are not mutually exclusive

• Determine $P(A)$, $P(B)$ and $P(A\text{ and }B)$.
• Add $P(A)$ and $P(B)$.
• Subtract the overlap:

$$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B).$$

The subtraction prevents outcomes in both events from being counted twice.

Using independent events

• Confirm that the outcome of one event does not affect the other.
• Multiply the probabilities:

$$P(A\text{ and }B)=P(A)\times P(B).$$

For a sequence represented by a tree diagram, multiply the probabilities along the relevant branch.

Using dependent events and tree diagrams

• Identify the first event and write its probabilities on the first set of branches.
• Determine how the first outcome affects the second event.
• Write the changed probabilities on the next branches.
• Multiply probabilities along each branch to find the probability of a sequence.
• Add the probabilities of alternative branches that produce the same final event.
• If items are selected without replacement, update the probabilities after each selection.

Calculating experimental probability and relative frequency

• Conduct or examine the experiment.
• Count the number of successful outcomes.
• Count the total number of trials.
• Calculate

$$\text{Experimental probability}=\frac{\text{number of successful trials}}{\text{total number of trials}}.$$

• For relative frequency, calculate

$$\text{Relative frequency}=\frac{\text{frequency}}{\text{total frequency}}.$$

Experimental probability is based on observed data rather than solely on a mathematical model.

Calculating expected frequency

• Identify the probability of the event.
• Identify the number of trials.
• Multiply:

$$\text{Expected frequency}=\text{probability}\times \text{number of trials}.$$

The expected frequency is a prediction; it does not guarantee the actual observed frequency.

Comparing theoretical and experimental probability

• Calculate the theoretical probability from equally likely outcomes or a mathematical model.
• Calculate the experimental probability from observed results.
• Compare the two values.
• Recognise that increasing the number of trials often brings experimental probability closer to theoretical probability, although variation remains possible.

Selecting a sample

• Identify the population being studied.
• Select a sample from that population.
• Consider whether the sample is sufficiently large.
• Check whether it represents the relevant subgroups of the population.
• Identify possible sources of bias.
• Choose an appropriate method:

- Random sampling: every population member has an equal chance of selection. - Systematic sampling: members are selected using a regular interval. - Stratified sampling: the sample reflects important subgroups in the population.

• Use the sample to draw conclusions about the population only when its representativeness is justified.

Where It Goes Wrong

• Treating a probability between 0 and 1 as a guarantee of what will happen in one individual trial.
• Forgetting that all probabilities in a complete sample space must add to 1, or accepting a probability outside $0\le P(A)\le 1$.
• Adding probabilities for events that are not mutually exclusive without subtracting $P(A\text{ and }B)$.
• Multiplying probabilities without checking whether the events are independent.
• Keeping probabilities unchanged when items are selected without replacement, even though the next probability may be dependent on the first outcome.
• Treating expected frequency as the actual number of occurrences, or ignoring possible bias caused by a small, one-subgroup, voluntary-response or leading-question sample.

What Gets Asked

This material supports questions requiring students to:

• Place events on the probability scale and identify impossible and certain events.
• Construct or interpret a sample space.
• Calculate theoretical probability from equally likely outcomes.
• Use complements to find the probability that an event does not happen.
• Decide whether events are mutually exclusive, independent or dependent.
• Apply addition and multiplication rules for combined events.
• Complete and interpret tree diagrams, including cases involving selection without replacement.
• Calculate experimental probability, relative frequency and expected frequency.
• Compare experimental and theoretical probability and explain the effect of increasing the number of trials.
• Identify the population, sample and possible bias in a sampling method.
• Select or justify random, systematic or stratified sampling.
• Evaluate whether a sample is sufficiently large and representative to support reliable conclusions.